An isometric extensor of metrics

Abstract

In this paper, for a metrizable space $Z$, we consider the space of metrics that generate the same topology of $Z$, and that space of metrics is equipped with the supremum metrics. For a metrizable space $X$ and a closed subset $A$ of it, we construct a map $E$ from the space of metrics on $A$ into the space of metrics on $X$ such that $E$ is an extension of metrics and preserves the supremum metrics between metrics.

36 pages. I have fixed gaps on topologies of Wasserstein spaces on arbitrary metric spaces. I have also provided a general observation on topologies on spaces of measurable functions using Lusin's theorem